The news was announced by USA Wrestling hours before the competition was due to begin in Fort Worth, Texas.
Cox is a reigning two-time world champion at the non-Olympic weight of 92kg, and had not lost once in his last 20 matches stretching back to 2018.
In February 2020, he shocked the wrestling world when he announced that he would be moving up a weight category for the U.S. Trials, rather than go back down to his Rio Olympic weight of 86kg.
The Missouri native had been given a bye into Friday's challenge tournament semi-finals, and his expected clash with reigning Olympic champion Kyle Snyder for a place in Japan was one of the most keenly-anticipated of the event.
The only way American wrestlers can qualify for the Olympics is to win the USA Trials, meaning Cox's hopes of improving on his bronze-medal finish at Rio 2016 are over for now.